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2) unknown coefficients. 1 after the unknowns, we get the entries for the design matrix A that links observations i and unknowns c¯n,m , s¯n,m : 27 28 CHAPTER 5. 3) s¯ GM sin (mli ) P¯n,m (cos Ji ) . 5) b = AT y. 6) The estimated coefficients are then obtained by computing xˆ = N−1 b. 1 holds only for points on the sphere with radius R. 8) where r is the radius of the point. 7. 2. 7. The inversion of the normal equation matrix N is usually replaced by a decomposition of N and solving for the unknown coefficients.

1 Description The previous chapter presented the “direct method” for solving linear equation systems, in this case for spherical harmonic analysis. As has been said before, its drawbacks are the memory requirements and the time required for the matrix multiplication N = AT A. To circumvent these problems, iterative solving methods have been developed, with the “conjugate gradients” probably being the most popular method. With this method, it is not necessary to fully build N. The computation only requires vector-vector operations, so very little memory is required.

1 Timing Before parallelising a program, we first need to know which parts of a program need the most computation time. It does not make sense to spend a lot of time and effort parallelising program parts that contribute only very little to the total runtime. When timing a program, there are three different time spans to be considered: • wall time: The time span a “clock on the wall” would measure, which is the time elapsed between start and completion of the program. This is usually the time to be minimised.

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